Combinations

Definition

Combinations are methods for calculating the number of ways to select a subset of items from a larger set where the order of selection does not matter.

Why It Matters

Combinatorics is the math of “how many.” Combinations represent grouping without structure, vital for evaluating hands of cards, creating committees, or selecting subsets in experimental design. Knowing how to strip order out of your calculations prevents combinatorial explosion.

Core Concepts

• Combinations (Order Doesn’t Matter):
  • Formula: $C(n, r) = \binom{n}{r} = \frac{n!}{r!(n-r)!}$
    • How to read: “The number of combinations of n items taken r at a time, or n choose r, is equal to n factorial divided by the product of r factorial and the quantity n minus r factorial.”
    • Meaning / when to use: Unordered selections—divide permutations by $r!$ to ignore order.

Connected Concepts

• Permutations
• The Binomial Theorem
• Multiplication Principle
• Addition Principle
• Inclusion-Exclusion Principle
• Probability Models: Equally Likely Outcomes